proof: Limit of self-super-roots is e^1/e. TPID 6 - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: proof: Limit of self-super-roots is e^1/e. TPID 6 ( /showthread.php?tid=362) |

proof: Limit of self-super-roots is e^1/e. TPID 6 - bo198214 - 10/07/2009
In reply to http://math.eretrandre.org/tetrationforum/showthread.php?tid=162&pid=4073#pid4073 First it is easy to see that for : ( is the lower fixed point of ) Hence for we have for all : (*) We also know that for , quite fast, particularly for each there is an such that for all : (**) . Now we lead proof by contradiction, suppose that where . Then there must be a subsequence and such that this subsequence stays always more than apart from : . I.e. there is and such that either or . By (*) and (**) we have such that for all : and . As is monotone increasing for we have also and . This particularly means and hence none of the can be the self superroot, in contradiction to our assumption. RE: proof: Limit of self-super-roots is e^1/e. TPID 6 - andydude - 10/07/2009
Wow! Very nice! You make it seem so easy. I've been working on that one for while, ever since the xsrtx thread. RE: proof: Limit of self-super-roots is e^1/e. TPID 6 - Base-Acid Tetration - 07/10/2010
The same method of proof could possibly be used to easily prove that, possibly for all k>4, limit of self-hyper-k-root(x) as x -> infinity = (defined as the largest real x such that , i.e. where the maximum of self-hyper-(k-1)-root function occurs; let's establish this notation); yeah I know, I only substituted the pentation-analogues into the proof and quickly checked. RE: proof: Limit of self-super-roots is e^1/e. TPID 6 - bo198214 - 07/10/2010
(07/10/2010, 05:19 AM)Base-Acid Tetration Wrote: The same method of proof could possibly be used to easily prove that, possibly for all k>4 The thing is: to define the hyper k-self-root you need a hyper (k-1) operation defined on the reals. And we still have several methods of doing this without equality proofs. |